Problem 1835

Let $k$ be the number of digits of $N$ and let $N = \sum_{i=0}^{k-1} 10^{i} \cdot d_{i}$ where $d_{i}$ are digits. If $M$ is obtained by swapping the $i$-th and the $j$-th digits, then
$$M - N = (10^{j} - 10^{i}) \cdot (d_{i} - d_{j}) .$$
Therefore $13 \mid M - N$ implies that $d_{i} = d_{j}$ or $6 \mid i - j$. Hence, a number $N$ satisfies the property given in the problem if and only if $13 \mid N$, $d_{i} \neq d_{j}$ for all $i \neq j$ and no swap between $i$-th and $j$-th digits are possible if $6 \mid i - j$.

- If $k \geq 8$, one can swap the 0th and the 6th digits, hence $N$ does not satisfy the property.
- If $k = 7$, one can swap the 0th and the 6th digits unless the 0th digit is 0, but if the 0th digit is 0, then no swap is possible.

Hence, $N$ satisfies the property if and only if $13 \mid N$, all the digits are distinct and the 0th digit is 0. Therefore, $N \leq 9876540$.

Considering all multiples of 13 that end with 0 below this level, one finds $N \in \{9876490, 9876360, 9876230, \ldots\}$. Since the two greatest elements of this set do not have all distinct digits, the greatest integer $N = 9876230$.